We study the problem of learning to rank from pairwise preferences, and solve
a long-standing open problem that has led to development of many heuristics but
no provable results for our particular problem. Given a set V of n
elements, we wish to linearly order them given pairwise preference labels. A
pairwise preference label is obtained as a response, typically from a human, to
the question "which if preferred, u or v?fortwoelementsu,v\in V.Weassumepossiblenonβtransitivityparadoxeswhichmayarisenaturallyduetohumanmistakesorirrationality.Thegoalistolinearlyordertheelementsfromthemostpreferredtotheleastpreferred,whiledisagreeingwithasfewpairwisepreferencelabelsaspossible.Ourperformanceismeasuredbytwoparameters:Thelossandthequerycomplexity(numberofpairwisepreferencelabelsweobtain).Thisisatypicallearningproblem,withtheexceptionthatthespacefromwhichthepairwisepreferencesisdrawnisfinite,consistingof{n\choose 2}$ possibilities only. We present an active learning algorithm for
this problem, with query bounds significantly beating general (non active)
bounds for the same error guarantee, while almost achieving the information
theoretical lower bound. Our main construct is a decomposition of the input
s.t. (i) each block incurs high loss at optimum, and (ii) the optimal solution
respecting the decomposition is not much worse than the true opt. The
decomposition is done by adapting a recent result by Kenyon and Schudy for a
related combinatorial optimization problem to the query efficient setting. We
thus settle an open problem posed by learning-to-rank theoreticians and
practitioners: What is a provably correct way to sample preference labels? To
further show the power and practicality of our solution, we show how to use it
in concert with an SVM relaxation.Comment: Fixed a tiny error in theorem 3.1 statemen